Did the Chabad Rebbe die?
For those who say the Chabad Rebbe is dead, they have some very compelling evidence. Namely, they have satisfied "habeas corpus". We have a body, a gravesite, no controversy surrounding the whereabouts of the body of the Chabad Rebbe. He has not been seen nor heard from (at least verifiably) since his assumed death in 1994. I didn't even need DNA or forensics for this.
For those who say that the CR lives, there are two categories. There are those who say that the CR lives on spiritually as a Tzadik, as all tzadikim. They would quote kabalah, that a tzadik is more alive after he leaves the world. They would say that a tzadik lives on in his teachings which are his true self, along the lines of the Tanya's explanation of the difference between a tzadik and a benoni. I am at a loss as to what they mean by this. How is the CR different in this than any of the multitude of tzadikim before him? For instance, if Yosef Karo, Moshe Isserles, Ran, Rif etc. are tzadikim, are they dead or alive in this view? How about the other Chabad Rebbes? Are they all alive? If you answer that they all are indeed alive, you have redefined alive in a consistent manner. You have conceded though that there is nothing special about the CR being "alive", except that you are saying that he is a tzadik. The physical evidence presents no problem. Kol HaKavod.
Then, there is the second category. There are those who say that the CR's death is different than other tzadikim, in that it is physical life. The physical evidence is problematic, so they answer with two parallels. During the Golden Calf experience, the people were shown that Moshe had died, in order to entice them to rebel. This is fairly weak, since they didn't have an actual body in that case and Moshe was indeed still alive. The information was deceptive, so the case is not a real parallel. So, they answer that Yaakov did not die. As Rashi brings from Aggada, even though they presented physical evidence that he had been buried and prepared as a corpse (I do not like the translation "embalm", for Chanata), nonetheless, he was only considered to have died. This explains the use of the word "Vayigva" rather than the more direct "vaYamas". In the case of Yaakov though, we have a pasuk that we must learn. Nontheless, we have a direct parallel. There is no telling what this really means, though. Would we say that, if his wife were alive, she could marry? If we can assign an aggada to an actual person, how do we know that it won't apply to others? What marked the CR as having this medrash apply to him, whereas others are distinguished as not having it apply to them? The question is immensely important. How can we ever permit any widow to remarry if undisputed appearance of death, burial and even preparing the corpse are not acceptable as evidence? One might answer that halacha uses assumption rather than proof, but even what the witnesses are expected to return in the case of an agunah is the evidence of appearance of death, based on the assumption that this is enough to assume death. This argument turns that very assumption on its head and invalidates halachas on which rests "misa v'kares". Another obvious problem is that when someone is dead - to all appearances - and one insists that that person is still alive, what exactly are they saying? In more clinical terms, what properties are different between the living and dead? We must determine this before we can assert that the difference does indeed exist. I deeply fear that most of the people who hold this view of the CR have not even thought it through to that extent.
So all in all, the primary question is not whether the CR died, but what do adherents mean when they say that he is still alive?